On that matter, last February's presidential election here was open to expats too, with ballots open in embassies/community centres in countries with significant Cypriot expat population (Austria, Belgium, Bulgaria, France, Germany, Greece, USA, UK, Qatar, Bahrain, Hungary, Saudi Arabia). Among them there must be holders of dual citizenship, eg Cypriot Brits (or British Cypriots I guess...
Charlies in any case) generally acquire the UK citizenship.
But they make only 2,6% of the electorate according to my quick maths. They can't influence the elections in a way that negates the vote of the population residing in Cyprus.
Here the main concern is how difficult the path to citizenship is
even for people born in the Republic,
even if they have one Cypriot parent. Not only they don't get to vote in the country they live in, but when they turn 18, they try to deport them in their "native countries", even though those kids have never been in those countries and don't even speak the language
Obviously, I don't know what's the case with Hungary. You do hear of countries with more citizens abroad than within its borders.